Cover potatoes with 1 inch of cold water in a large pot or Dutch oven. Bring to a boil, then reduce heat to medium. Cook until potatoes are tender and a knife inserted in center of a potato meets no resistance, 25 to 30 minutes.

Drain and let cool slightly. When potatoes are cool enough to handle, chop into 1-inch pieces and place in a large bowl. Toss potatoes with soup mix, vinegar, and pepper to taste. Cover bowl with plastic wrap and refrigerate until cool, 20 to 25 minutes.

Toss potatoes with celery, onion, mustard, mayonnaise, and lemon zest. Fold in arugula and parsley. Taste and adjust seasoning, if necessary. Serve immediately, or chill, covered, until ready to use.